Axiom ax-wl-cleq 38008

Description: Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong.

This text should be read as an exploration rather than as a definite statement, open to doubt, alternatives, and reinterpretation.

At the point where df-cleq 2757 is introduced, the foundations of set theory are being established through the notion of a class. A central property of classes is what elements, expressed by the membership operator ∈, belong to them . Quantification (∀𝑥) applies only to objects that a variable of kind setvar can represent. These objects will henceforth be called sets. Some classes may not be sets; these are called proper classes. It remains open at this stage whether membership can involve them.

The formula given in df-cleq 2757 (restated below) asserts that two classes are equal if and only if they have exactly the same sets as elements. If proper classes are also admitted as elements, then two equal classes could still differ by such elements, potentially violating Leibniz's Law. A future axiom df-clel 2840 addresses this issue; df-cleq 2757 alone does not.

**Primitive connectives and class builders**

Specially crafted primitive operators on classes or class builders could introduce properties of classes beyond membership, not reflected in the formula here. This again risks violating Leibniz's Law. Therefore, the introduction of any future primitive operator or class builder must include a conservativity check to ensure consistency with Leibniz's Law.

**This axiom covers only some principles of equality**

The notion of equality expressed in this axiom does not automatically coincide with the general notion of equality. Some principles are, however, already captured: Equality is shown to be an equivalence relation, covering transivity (eqtr 2785), reflexivity (eqid 2765) and symmetry (eqcom 2772). It also yields the class-level version of ax-ext 2737 (the backward direction of df-cleq 2757) holds.

If we assume 𝑥 = 𝐴 holds, then substituting the free set variable 𝑦 with 𝐴 in ax6ev 1992 and ax12v2 2217 yields provable theorems (see wl-isseteq 38011, and wl-ax12v2cl 38012). However, a bound variable cannot be replaced with a class variable, since quantification over classes is not permitted. Taken together with the results from the previous paragraph, this shows that a class variable equal to a set behaves the same as a set variable, provided it is not quantified.

**Conservativity**

Moreover, this axiom is already partly derivable if all class variables are replaced by variables of type "setvar". In that case, the statement reduces to an instance of axextb 2740. This shows that the class builder cv 1562 is consistent with this axiom.

**Eliminable operator**

Finally, this axiom supports the idea that proper classes, and operators between them, should be eliminable, as required by ZF: It reduces equality to their membership properties. However, since the term 𝑥 ∈ 𝐴 is still undefined, elimination reduces equality to just something not yet clarified.

**Axiom vs Definition**

Up to this point, the only content involving class variables comes from the syntax definitions wceq 1563 and wcel 2145. Axioms are therefore required to progressively refine the semantics of classes until provable results coincide with our intended conception of set theory. This refinement process is explained in Step 4 of wl-cleq-2 38003.

From this perspective, df-cleq 2757 is in fact an axiom in disguise and would more appropriately be named ax-cleq.

At first glance, one might think that 𝐴 = 𝐵 is defined by the right-hand side of the biconditional. This would make 𝑥 ∈ 𝐴, i.e. membership of a set in a class, the more primitive concept, from which equality of classes could be derived. Such a viewpoint would be coherent if the properties of membership could be fully determined by other axioms. In my (WL's') opinion, however, the more direct and fruitful approach is not to construct class equality from membership, but to treat equality itself as axiomatic.

(Contributed by Wolf Lammen, 25-Aug-2025.)

Assertion

Ref	Expression
ax-wl-cleq	⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Detailed syntax breakdown of Axiom ax-wl-cleq

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3	1, 2	wceq 1563	. 2 wff 𝐴 = 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1562	. . . . 5 class 𝑥
6	5, 1	wcel 2145	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2145	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wb 209	. . 3 wff (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
9	8, 4	wal 1561	. 2 wff ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
10	3, 9	wb 209	1 wff (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

This axiom is referenced by: (None)